Optimal Control of Network-Coupled Subsystems: Spectral Decomposition and Low-Dimensional Solutions

Abstract

In this article, we investigate the optimal control of network-coupled subsystems with coupled dynamics and costs. The dynamics coupling may be represented by the adjacency matrix, the Laplacian matrix, or any other symmetric matrix corresponding to an underlying weighted undirected graph. Cost couplings are represented by two coupling matrices which have the same eigenvectors as the coupling matrix in the dynamics. We use the spectral decomposition of these three coupling matrices to decompose the overall system into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$({L+1})$</tex-math></inline-formula> systems with decoupled dynamics and cost, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> is the number of linearly independent eigendirections associated with nonzero eigenvalue triples of the three coupling matrices. Furthermore, the optimal control input at each subsystem can be computed by solving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$({L_\text{dist}+1})$</tex-math></inline-formula> decoupled Riccati equations, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_\text{dist}\,(L_\text{dist}\leq L)$</tex-math></inline-formula> is the number of distinct nonzero eigenvalue triples of the three coupling matrices. A salient feature of the result is that, given the spectral decompositions of the couplings, the solution complexity does not directly depend on the number of subsystems. Therefore, the proposed solution framework provides a scalable method for synthesizing and implementing optimal control laws for large-scale network-coupled subsystems.